Image (and Video) Coding and Processing Lecture 2: Basic Filtering

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# Image (and Video) Coding and Processing Lecture 2: Basic Filtering - PowerPoint PPT Presentation

Image (and Video) Coding and Processing Lecture 2: Basic Filtering. Wade Trappe. Lecture Overview. Today’s lecture will focus on: Review of 1-D Signals Multidimensional signals Fourier analysis Multidimensional Z-transforms Multidimensional Filters. 1-D Discrete Time Signals.

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### Image (and Video) Coding and ProcessingLecture 2: Basic Filtering

Lecture Overview
• Today’s lecture will focus on:
• Review of 1-D Signals
• Multidimensional signals
• Fourier analysis
• Multidimensional Z-transforms
• Multidimensional Filters
1-D Discrete Time Signals
• A one-dimensional discrete time signal is a function x(n)
• The Z-transform of x(n) is given by
• The Z-transform is not guaranteed to exist because the summation may not converge for arbitrary values of z.
• The region where the summation converges is the Region of Convergence
• Example: If U(n) is the unit step sequence, an x(n)=anU(n), then
1-D Discrete Time Signals, pg. 2
• If the ROC includes the unit circle, then there is a discrete Fourier transform (found by evaluating at z=ejw):
• The inverse transform is given by:
• Observe: The DFT is defined in terms of radians! It is therefore periodic with period 2p!
• Parseval/Plancherel Relationship:
1-D Discrete Time Signals, pg. 3
• Discrete time linear, time-invariant systems are characterized by the impulse response h(n), which define the relationship between input x(n) and output y(n)
• This is convolution, and is expressed in the transform domain as:
• Causality: A discrete-time system is causal if the output at time n does not depend on any future values of the input sequence. This requires that
1-D Filters
• The impulse response for a system is also called the system’s transfer function.
• In general, transfer functions are of the form
• A system is a finite impulse response (FIR) system if H(z)=A(z), i.e. we can remove the denominator B(z)
• That is, the impulse response has a finite amount of terms.
• An infinite impulse response system is one where H(z) has an infinite amount of non-zero terms.
• Example of an IIR system:
1-D Filters, pg. 2
• A discrete-time system is said to be bounded input bounded output (BIBO stable), if every input sequence that is bounded produces an output sequence that is bounded.
• For LTI systems, BIBO stability is equivalent to
• Stability in terms of the poles of H(z):
• If H(z) is rational, and h(n) is causal, then stability is equivalent to all of the poles of H(z) lying inside of the unit circle
Sampling: From Continuity to Discrete
• The real world is a world of continuous (analog) signals, whether it is sound or light.
• To process signals we will need “sampled” discrete-time signals
• Analog signals xa(t) have Fourier transform pairs
• Let us define the sampled function x(n)=xa(nT). The Fourier transforms are related as:
• (Note: This is a good, little homework problem… will be assigned!)

0

2p/T

-4p/T

-2p/T

4p/T

Sampling: From Continuity to Discrete
• The effect of the sampling in the frequency domain is essentially
• Duplication of Xa(W) at intervals of 2p/T
• Pictorially, we have something like the following:
• Note: If the shifted copies overlap, then its “impossible” to recover the original signal from X(w).

1/T Xa(W)

Shifted Copies

Aliasing

Sampling: From Continuity to Discrete, pg. 2
• Aliasing occurs when there is overlap between the shifted copies
• To prevent aliasing, and ensure recoverability, we can apply an “anti-aliasing” filter to ensure there is no overlap.
• The overlap-free condition amounts to ensuring that
• If , then we say that xa(t) is W-bandlimited.
• As a consequence of the overlap-free condition, if we sample at a rate at least W, then we can avoid aliasing.
• This is, essentially, Shannon’s sampling theorem.
Multidimensional signals
• A D-dimensional signal xa(t0,t1,…,tD-1) is a function of D real variables.
• We will often denote this as xa(t), where the bold-faced t denotes the column vector t=[t0, t1, …, tD-1]T.
• The subscript “a” is just used to denote the analog signal. Later, we shall use the subscript “s” to denote the sampled signal, or no subscript at all.
• The Fourier transform of xa(t) is defined by
Multidimensional signals, pg. 2
• The Fourier transform is thus a scalar function of D variables.
• The Fourier transform is (in general) complex!
• The Inverse Fourier transform of Xa(W) is defined by
• Define the column vector of frequencies
• We get these relationships

Note the difference that D-dimensions introduces compared to 1-d Fourier Transform!

Example 2D Fourier Transform

Note: Ringing artifacts, just like 1-D case when we Fourier Transform a square wave

Image example from Gonzalez-Woods 2/e online slides.

W1

W1

W0

W0

Bandlimited Signals
• The notion of a bandlimited multidimensional signal is a straight-forward extension of the one-dimensional case:
• xa(t) is bandlimited if Xa(W) is zero everywhere except over a region with finite area.

Bandlimited

Not Bandlimited

Multidimensional Sampled Signals
• We will use n=[n0,n1,…,nD-1]T to denote an arbitrary D-dimensional vector of integer values
• A signal x(n) is just a function of D integer values
• The Fourier transform of x(n) and the inverse transform are given by
• Key point: X(w) is periodic in each variable wi with period 2p
Multidimensional Z transform
• The Z transform of x(n) is
• Plugging in gives X(w).
• We will often use the notation
• This notation will be useful later as it allows us to represent things in a way similar to the 1-dimensional Z-transform
Properties of Fourier and Z transforms
• Linearity
• Shift:

Hence, the multidimensional Z-transform is analogous to the one-dimensional delay operator

• Convolution:
Multidimensional Filters
• The basic scenario for multidimensional digital filters is:
• Convolution:
• Here, the transfer function is
• If x(n) has finite support, then y(n) will generally have larger support than x(n)

y(n)

x(n)

H(z)

w1

w1

w1

w0

w0

w0

Multidimensional Filter Response
• Just as in 1-D, the filter H can be characterized in terms of its frequency response.
• In this case, the frequency response is

Rectangular Lowpass

Diamond Lowpass

Circular Lowpass

w1

w1

w0

w0

Multidimensional Filters
• Multidimensional filters can be built by applying 1-D filters to each dimension separately
• These types of filters are separable.
• A separable filter is one for which the frequency response can be represented as:

Rectangular Lowpass

Not Separable

2-D Convolution, by hand…
• Rotate the impulse response array h(  ,  ) around the original by 180 degree
• Shift by (m, n) and overlay on the input array x(m’,n’)
• Sum up the element-wise product of the above two arrays
• The result is the output value at location (m, n)

From Jain’s book Example 2.1

For Next Time…
• Next time we will focus on multidimensional sampling.
• This lecture will be a blackboard/whiteboard style lecture.
• To prepare, read paper provided on website, and the discussion on lattices in the textbook